How can you rewrite $(z \leq-1.75)$ in order to find the answer?
A. $P (z \leq 1.75)$
B. $P (z \geq-1.75)$
C. $1-P(z \leq-1.75)$
D. $1-P(z \leq 1.75)$
Answer (2)
The problem requires rewriting P ( z ≤ − 1.75 ) using standard normal distribution properties.
Apply the symmetry property: P ( z ≤ − a ) = P ( z g e a ) .
Use the total probability rule: P ( z g e a ) = 1 − P ( z ≤ a ) .
Combine these to find the equivalent expression: P ( z ≤ − 1.75 ) = 1 − P ( z ≤ 1.75 ) .
The final answer is 1 − P ( z ≤ 1.75 )
Explanation
Understand the problem We are given the probability P ( z ≤ − 1.75 ) and asked to rewrite it using the provided options. The options are: P ( z ≤ 1.75 ) P ( z g e − 1.75 ) 1 − P ( z ≤ − 1.75 ) 1 − P ( z ≤ 1.75 ) We need to find an equivalent expression to P ( z ≤ − 1.75 ) among the given choices.
Symmetry of the standard normal distribution Let's recall some properties of the standard normal distribution. The standard normal distribution is symmetric around 0. This means that the probability of a value being less than or equal to − a is the same as the probability of a value being greater than or equal to a . In mathematical terms:
P ( z ≤ − a ) = P ( z g e a )
Total probability and its implications Also, we know that the total probability under the standard normal curve is 1. Therefore, the probability of a value being less than or equal to a plus the probability of a value being greater than a is equal to 1:
a) = 1"> P ( z ≤ a ) + P ( z > a ) = 1
Since the standard normal distribution is continuous, we can approximate a)"> P ( z > a ) as P ( z g e a ) . Thus,
P ( z ≤ a ) + P ( z g e a ) = 1
Rearranging this, we get:
P ( z g e a ) = 1 − P ( z ≤ a )
Applying symmetry to the problem Now, let's apply these properties to our problem. We want to rewrite P ( z ≤ − 1.75 ) . Using the symmetry property, we have:
P ( z ≤ − 1.75 ) = P ( z g e 1.75 )
Applying total probability to the problem Using the total probability property, we can rewrite P ( z g e 1.75 ) as:
P ( z g e 1.75 ) = 1 − P ( z ≤ 1.75 )
Therefore, we have:
P ( z ≤ − 1.75 ) = 1 − P ( z ≤ 1.75 )
Final Answer So, the equivalent expression to P ( z ≤ − 1.75 ) is 1 − P ( z ≤ 1.75 ) .
Examples
Understanding probabilities related to the standard normal distribution is crucial in many fields. For example, in quality control, it helps determine the likelihood of a product falling within acceptable quality ranges. In finance, it's used to assess the risk associated with investments. In medical research, it helps determine the significance of experimental results. The ability to manipulate and rewrite probabilities, as we did here, allows us to use statistical tables and software more effectively to solve real-world problems.
The expression P ( z ≤ − 1.75 ) can be rewritten as 1 − P ( z ≤ 1.75 ) using properties of the standard normal distribution. The symmetry of the normal curve allows us to relate probabilities of negative and positive values. Therefore, the correct option is D: 1 − P ( z ≤ 1.75 ) .
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